Obtaining adjusted prevalence ratios
from logistic regression models
in cross-sectional studies
Obtendo razões de chance prevalentes
de modelos de regressão logística
em estudos transversais
La obtención de las prevalencias ajustadas a
partir de los modelos de regresión logística
en los estudios transversales
1 Programa de Computação Científica, Fundação Oswaldo Cruz, Rio de Janeiro, Brasil.
2 Instituto de Pesquisa Clínica Evandro Chagas, Fundação Oswaldo Cruz, Rio de Janeiro, Brasil.
3 Departamento de Matemática e Estatística, Universidade Federal do Estado do Rio de Janeiro, Rio de Janeiro, Brasil.
Correspondência L. S. Bastos
Programa de Computação Científica, Fundação Oswaldo Cruz. Av. Brasil 4365, Rio de Janeiro, RJ
21040-360, Brasil. lsbastos@fiocruz.br
Leonardo Soares Bastos 1
Raquel de Vasconcellos Carvalhaes de Oliveira 2 Luciane de Souza Velasque 3
Abstract
In the last decades, the use of the epidemiologi-cal prevalence ratio (PR) instead of the odds ra-tio has been debated as a measure of associara-tion in cross-sectional studies. This article addresses the main difficulties in the use of statistical models for the calculation of PR: convergence problems, availability of tools and inappropri-ate assumptions. We implement the direct ap-proach to estimate the PR from binary regression models based on two methods proposed by Wil-cosky & Chambless and compare with different methods. We used three examples and compared the crude and adjusted estimate of PR, with the estimates obtained by use of log-binomial, Pois-son regression and the prevalence odds ratio (POR). PRs obtained from the direct approach resulted in values close enough to those obtained by log-binomial and Poisson, while the POR overestimated the PR. The model implemented here showed the following advantages: no nu-merical instability; assumes adequate probabil-ity distribution and, is available through the R statistical package.
Prevalence Ratio; Logistic Models; Cross-Sectional Studies
Resumo
Nas últimas décadas, tem sido discutido o uso da razão de prevalência (RP) ao invés da razão de chance como a medida de associação a ser estimada em estudos transversais. Discute-se as principais dificuldades no uso de modelos estatísticos para o cálculo da RP: problemas de convergência, disponibilidade de ferramentas e pressupostos não apropriados. O objetivo deste estudo é implementar uma abordagem direta para estimar a RP com base em modelos logísti-cos binários baseados em dois métodos propos-tos por Wilcosky & Chamblers, e comparar com outros métodos. Utilizou-se três exemplos e com-parou-se as estimativas bruta e ajustada da RP obtidas pela função com as estimativas obtidas pelos modelos log-binomial, Poisson e razão de chance prevalente (RCP). As RP da abordagem proposta resultaram em valores próximos aos obtidos pelos modelos log-binomial e Poisson, e a RCP superestimou a RP. O modelo aqui im-plementado apresentou as seguintes vantagens: não apresenta instabilidade numérica; assume a distribuição de probabilidades adequada; e está disponível no programa estatístico R.
Introduction
Over the past few decades, several authors
1,2,3,4,5,6,7 have tried to determine the most
ap-propriate association measure to be used in cross-sectional studies. The consensus is that the prevalence odds ratio (POR) is only a good ap-proximation of the prevalence ratio (PR) when the event of interest is rare 8. Logistic regression
is the most popular statistical model used in es-timating POR due to ease of interpretation and computational implementation. However, when the choice of association measure is the PR, and the event of interest is not rare, this model pro-duces poor estimates. In such cases, several au-thors have proposed alternatives to logistic re-gression models to estimate the true PR.
Lee & Chia 9 were the first authors to use Cox
models with Breslow’s modification (Breslow-Cox model) to estimate prevalence ratios, but in the study, standard errors and, consequently, confidence intervals, were not calculated correct-ly. The correction for standard errors obtained by Cox models had already been proposed 10,
but were not considered. Barros & Hirakata 5
used the fact that Breslow-Cox and Poisson mod-els estimate the same effects 11 and used
Pois-son regression models with robust variance to estimate the PR. Zou 12 published a simulation
study demonstrating the reliability of the Poisson model with robust variance to estimate PR in 2 by 2 tables. The main issue with using a Poisson model to estimate PR is the misuse of a specific counting probability distribution to describe a response variable that is dichotomous (presence or absence of an outcome).
Skov et al. 4 used a generalized linear model
with a binomial distribution and a log link (log-binomial model) to directly estimate PR 13.
Al-though this model allows for directly estimating the PR and assumes a probability distribution that agrees with the type of the response vari-able, the lack of convergence in the presence of continuous variables remains a problem. To solve this issue, Deddens et al. 6 introduced the
COPY method for finding an approximation to the MLE when the log-binomial model fails to converge. Due to the convergence problem of the log-binomial model, Schouten et al. 14 proposed
a simple data manipulation that allows for the use of logistic regression to obtain the PR. It con-sists in modifying the data set by duplicating the lines where the event occurs and replacing the outcome from event to non-event 14,15,16.
Another approach – proposed by Wilcosky & Chambless 17, using the conditional and
mar-ginal methods 18 – involves a direct adjustment
of epidemiological measures through binary
re-gression. An advantage of these methods is that they assume a probability distribution for a vari-able with a binomial response, matching the na-ture of the observed response variable in cross-sectional studies. We find one article 19 that uses
the Wilcosky & Chambless 17 method to estimate
PR, but it did not mention the software imple-mentation. Recently, R Core Team developed a software package in R (R package version 1.2; The R Foundation for Statistical Computing, Viena, Austria; http://www.r-project.org) for estimating marginal and conditional PRs and confidence in-tervals via bootstrap and delta methods, but they have yet to publish a scientific article explaining the details of this package and the differences be-tween the methods it utilizes to estimate PR.
In this article, we use a direct approach to estimate the prevalence ratio from binary re-gression models based on methods proposed by Wilcosky & Chambless 17, and we compare the
results to those obtained through different meth-ods presented in the literature. Three different data sets are used to illustrate our study.
Methods
Based on the approach proposed by Wilcosky & Chambless 17, we use real and simulated data to
compare PR estimates obtained by the marginal and conditional methods. Those estimates are also compared with the estimates obtained by the binomial, log-binomial and robust Poisson/ Cox models.
Using a logistic model, it is straightforward to estimate the probability of occurrence of a disease (denominated prevalence in transver-sal studies) adjusted for two or more variables. Suppose, for example, that one has information about the diabetes status (1: yes/0: no), age (con-tinuous) and obesity status (1: yes/0: no) of a defined population. With this information, one can obtain the adjusted probability of diabetes through the following equation:
(Equation 1),
where P is the probability that DIABETES = 1, β0,
β1, β2 are regression coefficients estimated from
the data. Note that exp(β2) estimates the odds
ra-tio for diabetes in obese individuals compared to non-obese individuals, adjusted by age. Howev-er, if we are interested in obtaining the estimated PR for diabetes, adjusted by age, in obese and non-obese individuals, we can proceed in two ways, as described below:
preva-lence is calculated for each age value included in the dataset using Equation 1. The PR is the ratio between the average of the prevalences in each stratum. Wilcosky & Chambless 17 refer to this
es-timate as the marginal prevalence ratio (MPR); 2) Conditional method: in each stratum of the variable OBESITY, the diabetes prevalence is cal-culated using Equation 1, setting age as an aver-age value obtained from the dataset. Thus, the ratio of the two prevalences can be calculated. Wilcosky & Chambless 17 refer to this estimate as
the conditional prevalence ratio (CPR).
In the linear regression model, both methods estimate the same value. However, in the logistic regression model, we observed significant differ-ences between the estimates of the two methods when p is close to zero or one.
According to Lee & Chia 9, the marginal
meth-od provides an internally adjusted measure, mak-ing invalid any comparisons to external values of PR. With the conditional method, on the other hand, one can use default values as the average values of covariates, allowing for comparisons with other population studies that used the same default values. More details about the marginal and conditional methods can be found in Lee 20
and Wilcosky & Chambless 17.
Asymptotic confidence intervals for the con-ditional and marginal prevalence ratios were proposed by Flanders & Rhodes 21. The authors
also presented an SAS (SAS Inc., Cary, USA) script for estimating and calculating the intervals of the conditional and marginal prevalence ratio. To the best of our knowledge, this was the only imple-mentation of these measures to date.
The prevalence ratio estimation methods are illustrated in three different databases, all con-taining a binary outcome Y, a binary exposure X, and at least one control variable Z.
Application 1: the first database was a toy example with 1,000 simulated observations and one continuous control variable. In this example, we simulated 1,000 binary outcomes with a bi-nary exposure, X, and a continuous confound-ing variable, Z. The exposure was sampled from a Bernoulli distribution with probability 0.5, the confounding variable was sampled from a Nor-mal distribution with mean zero and unit vari-ance. The outcome was sampled from a Ber-noulli distribution with probabilities such that: the baseline prevalence was 20%; the conditional prevalence ratio for X at Z = 0 was equal to 2; and the conditional prevalence ratio for X at Z = 1 was equal to 1.919 (regression coefficient β2 = 0.20).
There were several possible values for the con-ditional prevalence ratio for X depending on Z.
Application 2: the second database referred to a cohort of 1,273 live births in 1993 in the city
of Pelotas, Rio Grande do Sul, Brazil, studied with the aim of linking sociodemographic factors and reproductive health, informed by the responsi-ble female, to the nutritional condition of their children after 4-5 years 5. The analysis
consid-ered underweight in 4-5 year old children (with a prevalence of 4.1%) as the outcome of inter-est, Y; previous hospitalization as the exposure, X; and birth weight (normal or low birth weight) as a control variable, Z. For this application, be-cause all variables are binary, we were able to calculate the prevalence ratio applying the Man-tel-Haenszel method, considered here to be the gold standard.
Application 3: the third database analyzed 703 sexually active, HIV-infected women, treated between 1996 and 2007 in Rio de Janeiro, Brazil, with no history of hysterectomy. The data was collected in order to assess factors associated with high-grade squamous intraepithelial lesions (HSIL), lesions that can develop into cancer of the cervix 22. Five variables pertaining to the
HIV-infected women were included in analysis: ence of HPV was the exposure variable, X; pres-ence of cervical cytological abnormalities was the outcome, Y; and the control variables, Z, were age, number of pregnancies and the time since the last gynecological examination. Variables X and Y were binary variables and the others were continuous variables. The prevalence of the out-come was 4.1%.
Adjusted prevalence ratios and prevalence odds ratios were estimated by several different methods. Prevalence ratios were estimated by robust Poisson and log-binomial models, and by the conditional and marginal methods pro-posed by Wilcosky & Chambless 17. POR were
also calculated using the usual logistic regres-sion model.
The different methods to obtain prevalence ratios were coded in R (The R Foundation for Statistical Computing, Vienna, Austria; http:// www.r-project.org). An R function to estimate the conditional and marginal prevalence ratios, as proposed by Wilcosky & Chambless 17, is
avail-able (Figure 1).
Results
Application 1: toy example
Figure 1
R code developed to estimate the conditional and the marginal prevalence ratio, as proposed by Wilcosky & Chambless 17.
(continues)
##################################################### ### Prevalence ratio from logistic regression models
#####################################################
PR = function( model, type="marginal", level=0.95 ){ # model: output of a glm funtion family=binomial # type: "marginal", "conditional", "both"
p = model$rank-1 n = length(model$y)
xx=rep(0,model$rank-1)
modeloAux = update(model, x=T)
modeloAux2 = glm(modeloAux$y ~ modeloAux$x[,-1],
family=binomial(), weights=modeloAux$prior.weights)
labelCI = paste( (1 + c(-level,level))/2 * 100, "%")
auxMatrix = data.frame(PR = model$coefficients[-1], sd = NA) auxMatrix[, labelCI[1]] = NA
auxMatrix[, labelCI[2]] = NA
auxList = list(Conditional=auxMatrix, Marginal=auxMatrix)
# Condicional
xmean = apply(modeloAux$x,2, weighted.mean, w=modeloAux$prior.weights)
betaAux = modeloAux2$coefficients varbetaAux = vcov(model)
for(k in 1:p){ ##### Conditional XcondAuxXk = xmean
# Xk = x+1
XcondAuxXk[k+1] = xx[k]+1
etaxkC = drop(betaAux %*% XcondAuxXk)
# P(X_k = x+1)
PXk1C = 1/(1+exp(-etaxkC)) # \Delta P(X_k = x+1)
DeltaPXk1C = XcondAuxXk * PXk1C * (1 - PXk1C)
# X_k = x
XcondAuxXk[k+1] = xx[k]
etaxkC = drop(betaAux %*% XcondAuxXk)
# P(X_k = x)
PXk0C = 1/(1+exp(-etaxkC)) # \Delta P(X_k = x)
-1.96 to 1.96. The crude prevalence ratio (1.477) underestimates this range of the true prevalence ratio, whereas the crude and the adjusted preva-lence odds ratio (2.528 and 2.537) overestimate the true range (although their confidence
inter-vals overlap with some of the true range) (Table 1). The adjusted prevalence ratios are all very similar, and all provide reasonable estimates (Ta-ble 1). The estimates differ only in the second or third decimal places, with the smallest estimated Figure 1 (continued)
#### Conditional PR_Xk PRXkC = PXk1C / PXk0C
DeltaPRxkC = (DeltaPXk1C * PXk0C - DeltaPXk0C * PXk1C) / PXk0C^2 varPRxkC = crossprod(DeltaPRxkC, varbetaAux) %*% DeltaPRxkC
auxList$Conditional[k,] = c( PRXkC, sqrt(varPRxkC), PRXkC*exp( qnorm(c(.025,.975))*sqrt(varPRxkC)/PRXkC ))
##### Marginal
XmargAux = modeloAux$x
# Xk = 1
XmargAux[,(k+1)] = xx[k]+1
etaxkM = drop(XmargAux %*% betaAux)
PXk1M = mean(1/(1+exp(-etaxkM)))
auxVarMarg = exp(-etaxkM) / (1+exp(-etaxkM))^2
DeltaXk1M = 0 for(j in 1:n)
DeltaXk1M = DeltaXk1M + XmargAux[j,] * auxVarMarg[j] / n
# Xk = 0
XmargAux[,(k+1)] = xx[k]
etaxkM = drop(XmargAux %*% betaAux)
PXk0M = mean(1/(1+exp(-etaxkM)))
auxVarMarg = exp(-etaxkM) / (1+exp(-etaxkM))^2
DeltaXk0M = 0 for(j in 1:n)
DeltaXk0M = DeltaXk0M + XmargAux[j,] * auxVarMarg[j] / n
#### Marginal PR_Xk PRXkM = PXk1M / PXk0M
DeltaPRXkM = (DeltaXk1M * PXk0M - DeltaXk0M * PXk1M) / PXk0M^2 varPRXkM = crossprod(DeltaPRXkM, varbetaAux) %*% DeltaPRXkM
auxList$Marginal[k,] = c( PRXkM, sqrt(varPRXkM), PRXkM*exp( qnorm(c(.025,.975))*sqrt(varPRXkM)/PRXkM ))
}
switch(type,
marginal = return(auxList$Marginal), conditional = return(auxList$Conditional), both = return(auxList)
Table 1
Adjusted prevalence ratios and respective 95% confidence interval (95%CI) estimates in the analysis of toy data using Y as the outcome, X as the risk factor and the continuous covariate Z as a control factor.
Regression model (measure) Estimate 95%CI
Robust poisson (PR) 1.950 1.573, 2.416
Log-binomial (PR) 1.942 1.575, 2.418
Logistic regression (POR) 2.537 1.905, 3.398
Logistic regression (CPR) 1.956 1.578, 2.425
Logistic regression (MPR) 1.949 1.574, 2.414
CPR: conditional prevalence ratio; MPR: marginal prevalence ratio; POR: prevalence odds ratio; PR: prevalence ratio. Note: the true conditional prevalence ratio for X varies from 1.71 to 2.25 depending on the value of Z (-1.96 to 1.96),
value in the log-binomial model and the largest in the conditional prevalence ratio.
Application 2: underweight in 4-5 year-old children in Pelotas
Table 2 presents the adjusted prevalence ratio of the occurrence of underweight in 4-5 year-old children (outcome) by previous hospitalization (exposure) controlled by birth weight (normal or low birth weight).
Despite the low prevalence of the outcome (4.1%), a difference was observed of 0.169 be-tween the crude PR (2.902) and the crude POR (3.071) for previous hospitalization. According to the crude PR, there was a greater prevalence of underweight among children who were pre-viously hospitalized when compared with those without previous hospitalization. The adjusted prevalence ratios of the log-binomial, robust Poisson, marginal prevalence ratio and Mantel-Haenszel method presented similar estimates (2.481, 2.479, 2.460, and 2.483, respectively). The largest adjusted estimates were the POR (2.641) and the conditional prevalence ratio (2.532).
Application 3: cervical cytological abnormalities in HIV-infected women
Table 3 shows the influence of high risk HPV (ex-posure) in the occurrence of cervical cytological abnormalities in HIV-infected women, controlled by age, number of pregnancies and time since last gynecological examination. Despite the low prevalence, the crude POR (7.909) differed from the crude PR (7.360) by 0.6. Those women with high risk HPV presented 640% more cytological abnormalities. The adjusted POR was the high-est high-estimated value (7.990). The adjusted
preva-lence ratios obtained by the log-binomial, robust Poisson approach and the marginal prevalence ratio were very similar. The conditional preva-lence method led to a ratio up to 46% greater than those obtained from other adjusted methods.
Discussion
Difficulties in obtaining prevalence ratios in cross-sectional studies have been investigated by several authors in recent years. Several au-thors use strategies for indirect calculation of the PR using the Breslow-Cox and Poisson models (with and without robust variance), while others interpret the prevalence odds ratio obtained in logistic regression models as a prevalence ratio.
Lee & Chia 9 were the first authors to discuss
methods proposed for estimating the PR. Until then, most cross-sectional studies in health used the logistic regression model estimate (POR), since it has the advantage of adjusting for the confounding or modifying effects of other vari-ables. When the outcome is prevalent, however, the POR is a poor estimate of the prevalence ratio, overpredicting the PR by up to 27 times 23.
Regarding the estimation of the adjusted prevalence ratio, in our examples the log-bino-mial model, robust Poisson model, and marginal prevalence ratio provided similar estimates. The conditional prevalence ratio (CPR) differed from the other estimates but was still smaller than the adjusted POR. The CPR proposed by Wachold-er 13 is the prevalence ratio conditional on the
Table 2
Adjusted prevalence ratios and respective 95% confidence interval (95%CI) estimates in the analysis of the data using underweight as the outcome (Y), previous hospitalization as the risk factor (X) and birth weight as the control factor (Z).
Regression model (measure) Estimate 95%CI
Mantel-Haenszel (PR) 2.483 1.456, 4.235
Robust poisson (PR) 2.479 1.454, 4.226
Log-binomial (PR) 2.481 1.447, 4.226
Logistic regression (POR) 2.641 1.481, 4.671
Logistic regression (CPR) 2.532 1.471, 4.357
Logistic regression (MPR) 2.460 1.451, 4.171
CPR: conditional prevalence ratio; MPR: marginal prevalence ratio; POR: prevalence odds ratio; PR: prevalence ratio.
Table 3
Adjusted prevalence ratios and respective 95% confidence interval (95%CI) estimates in the analysis of cervical cytological abnormalities in HIV-infected women considering high risk HPV as the exposure variable (X), and controlling for three other variables (Z).
Regression model (measure) Estimate 95%CI
Robust poisson (PR) 7.123 2.489, 20.388
Log-binomial (PR) 7.192 2.849, 24.135
Logistic regression (POR) 7.990 3.029, 27.531
Logistic regression (CPR) 7.529 2.617; 21.665
Logistic regression (MPR) 7.118 2.518; 20.124
CPR: conditional prevalence ratio; MPR: marginal prevalence ratio; POR: prevalence odds ratio; PR: prevalence ratio. Note: Z variables = age, number of pregnancies, and time since last gynecological examination.
was estimated for those women who had high risk HPV (X = 1), based on their respective values for age, number of pregnancies, and time since last gynecological examination (Z variables), and the mean value of the prevalence was computed. Similar calculations were performed for those women who were not diagnosed with high risk HPV (X=0). More detailed information on condi-tional and marginal methods is well described in Wacholder 13 and Wilcosky & Chambless 17.
The main advantage of the log-binomial and robust Poisson models is that they are already implemented in most popular statistical pack-ages. The log-binomial has the disadvantage of not using a proper link function, leading to nu-merical instability in the estimation process and resulting in non-convergence issues. The COPY method 6 was proposed to achieve convergence
with the log-binomial model, but this method is
only available in SAS, which is a proprietary soft-ware. The robust Poisson model assumes that all the events in the database occurred at the same time. In addition, use of the Poisson distribution is not appropriate for modeling a binary outcome 6.
However, it is important to highlight that the likelihood of the Poisson model has been used only to obtain an estimation equation and not for the purpose of modeling a binary response variable. The Schouten et al. 14 approach can be
implemented easily on any statistical package, but in changing the database, it brings extra un-certainty that should be properly treated. The approach used by Wilcosky & Chambless 17,
un-like the log-binomial model, does not suffer from convergence problems.
tory variable. In Application 2, where all explana-tory variables were binary, the Mantel-Haenszel method was used as the “gold standard”. For this application, we found that the prevalence ratio estimated by the log-binomial model, Pois-son robust model and marginal prevalence ratio showed estimates similar to the one obtained by the Mantel-Haenszel method. We thus con-clude that, in this application, the equivalence of the models applied. In this paper we have not explored robust methods based on quasi-likeli-hood estimation 15.
In summary, we recommend the use of the direct approach proposed by Wilcosky &
Chamb-less 17 because it is suitable for a binary response
when using a variable binomial model, it has no convergence difficulties and it is now avail-able as a package for the open source statistical software R. The estimates of the marginal preva-lence ratio are similar to those of other methods, while the conditional prevalence ratio shows the prevalence ratio for an average person in the database. If one is interested in a particular set of control variables, one only need specify the values of those variables. The authors are devel-oping an R package with the Wilcosky & Chamb-less approach, which will be available along with this article.
Resumen
En las últimas décadas, se ha discutido el uso de la ra-zón de prevalencia (RP), en lugar del odds ratio como medida de asociación que se estima en estudios trans-versales. Se analizan las principales dificultades en el uso de modelos estadísticos para el cálculo de la RP: problemas de convergencia, disponibilidad de her-ramientas y supuestos no apropiados. El objetivo es realizar un enfoque directo para estimar la RP desde modelos logísticos binarios, basados en dos métodos propuestos por Wilcosky y Chamblers y compararlos con otros métodos. Se han utilizado 3 ejemplos y com-paramos las estimaciones crudas y ajustadas de RP con las estimaciones obtenidas por log-binomial, Poisson y odds ratio de prevalencia (ORP). Los RP obtenidos del enfoque directo dieron como resultado valores cercanos a los obtenidos mediante el log- binomial y de Poisson, mientras que la RCP sobreestimó la RP. El modelo que aquí se presenta implementó las siguientes ventajas: no presenta inestabilidad numérica, toma una distribuci-ón de probabilidad apropiada y está disponible en sof-tware estadístico libre R.
Razón de Prevalencias; Modelos Logísticos; Estudios Transversales
Contributors
L. S. Bastos participated in the drawing up of the R func-tion, literature revision, and article write up. R. V. C. Oli-veira and L. S. Velasque collaborated in the literature revision and article write up.
Acknowledgments
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Submitted on 08/Oct/2013
